Do you get dropped from your parents insurance the day you graduate from college or is there a grace period to allow you to get insurance on your own?

Usually when you turn 23, 24 or 25, depending on what the contract says.  
If you are on your parent's insurance and are going to college full time then you will be covered until you graduate. Once you graduate check with a local insurance agent to see about obtaining insurance for yourself the day after you graduate.  
You generally have a small grace period (30 days is typical) after you graduate from college assuming you did not already reach some other qualifying event like age as referenced above or failure to maintain full-time status. Also, if your parents insurance is individual coverage and not group coverage, you may have that same 30 days to convert to your own individual coverage on the same plan without any medical underwriting at all. Basically you just tell them to move you to your own plan. Insurance companies don't like to tell you this because they want the opportunity to decide if they want to cover you or not. Check your parent's plan carefully for this guaranteed coverage option.